Introduction
Functions are used to describe mathematical things and can be difficult to define. The basic definition of a function can be said to be – a collection of ordered pairs of things, where the first members are fundamentally different in the pairs.
A simple function can be as follows:
[{1, 2}, {2, 4}, {3, 6}, {4, 18}, {5, 10}]
The above function has five pairs where the first members are 1, 2, 3, 4 and 5.
Functions usually have alphabetical letter as their names. So if we term this function ‘f’, which is the most common letter used for functions, then it will be properly written as:
f(1) = 2, f(2) = 4, f(3) = 6, f(4) = 8, f(5) = 10
Here are two definitions to keep in mind:
The entire set of first numbers in the function is called a domain and the first members are called arguments. In this particular example, the domain has 5 numbers and the numbers 1, 2, 3, 4 and 5 are the arguments of the function.
The whole set of second numbers in the function is called the range and the second members are called the values. Going back to the above function, the range also has 5 numbers and the numbers 2, 4, 6, 8 and 10 are the values of the function.
As mentioned before, the standard naming of a function is f. Thus we can explain this function in a sentence as follows:
The value of the function (f) at argument 1 is 2, its value at argument 2 is 4, its value at argument 3 is 6, its value at argument 4 is 8 and its value at argument 5 is 10.
Therefore a function can also be defined as a set of assigned values (the second numbers) to arguments (the first numbers)
This can be expanded further to say that the condition is that the first member of every pair is different; therefore each argument of the domain of function ‘f’ gets an exclusive value in its range.
The linear function and its importance to calculus
The linear function is the basic and essential function, on which calculus is based upon. This is a function that has a straight line running through the domain of its graphs.
Such a line can be determined by two points that lie on it. Look at the function [a, f{a}], [b, f{b}]. You can pick an “a” and “b” in the domain and determine this line defined by the two values f{a} and f{b}.
Let’s look at the formula for such a function.
It is possible to determine the linear function for the two values mentioned above by using the following formula.
f{x} = [f{a} x – b/ a – b] + [f{b} x – a/ b – a ]
Effectively this means that the first term is 0 when x is equal to b, and it becomes f{a} when x equals a. The second term is 0 when x is equal to a and it becomes f{b} when x equals b.
Another important aspect of a linear function is its slope.
This is defined as the ratio of the change of function f between x = a and x = b the change in x between the two arguments. The y-intercept is the point at which the line passes the y-axis.
The intercept of the line on the y axis is also an essential part of the linear function.
As we have seen, a linear function can be defined one that has a graph with a straight line, and can be described by its slope and y-intercept.
Special linear functions are often useful and they all have an important and unique property – they all have linear functions whose y-intercepts go through the point 0. Their graphs pass through the origin of the x and y axes. They are aptly called homogenous linear functions, and they all share the same property which is:
Their value at any permutation of two arguments is equal to the same permutations of their values at those arguments.
This can be explained by the following formula:
F{ak + bc} = af{k} + bf{c}
The above property is called the “property of linearity”.
NOTE: not all linear functions have this property of linearity. The property implies that once you know the value of a linear function and any two distinct arguments, then you can find the value at any other point or pair of arguments. This is not always true.
Practical applications of the linear functions
There are several real life applications of calculus linear functions. Remember that this is the most basic function on which other functions are based upon. The function is applied in various fields, such as meteorology, pharmaceuticals, engineering, and a lot more.
Whenever you have to create a graph in a straight line, no matter what the slope or y-intercept is, you are applying this basic principle.
NOTE: One should not confuse linear functions in calculus to linear equations in algebra. They have different properties even if sometimes their graphs can be identical. You can find a graph for a linear equation of algebra having the same slope and y-intercept as a graph for linear function of calculus, but they do not represent the same properties.
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Conclusion
Starting off by understanding this basic formula of calculus will make it very easy for you to move on and understand the deeper functions or integration and differentiation. Calculus should not be a behemoth to be feared but a friend to be understood. Try out some basic exercises on the linear functions in calculus and you will get a better grip on the topic.
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Polynomials are algebraic expressions that have a higher degree than the standard x + 3 or y – 2. A polynomial refers to anything with a degree, or highest exponent, above 2, but usually means at least 4. Some of the polynomials you’ll see most often are cubic polynomials or expressions with a cube as their highest variable.
Often in higher math such as calculus, you’ll be asked to solve these expressions, However, if you’ve been working with higher math for any time, you know that the more complex that a math problem is, the more difficult its variables and solutions will be. One way, and often a mandatory step, in solving cubic polynomials is to factor the expression.
We’ll go over some ways to factor cubic polynomials, but first, let’s review what these are.
A polynomial is any expression that has multiple terms in it. Terms are variables or numbers such as x^{2}, 3x^{3}, and 5. In algebraic expressions, you don’t know what the variables represent so you can’t add them. You can only add or subtract terms that have the same degree and the same variable. For example, you could subtract 5x from 8x, but not from 8y.
Therefore, an important part of learning how to factor cubic polynomials in calculus and other forms of math is learning how to simplify.
The first thing to remember is that not all polynomials can be factored. Many if not most of them, can, but there are a few that have no roots at which x is equal to zero. However, most polynomials can be simplified into a single expression multiplied by a quadratic expression. For example, you might see (x^{2} – 2x +4)(x + 3).
It’s even possible that the quadratic equation can factor further, but we’ll get to that later. The first step to factoring a cubic polynomial in calculus is to use the factor theorem.
The factor theorem holds that if a polynomial p(x) is divided by ax – b and you have a remainder of 0 when it’s expressed as p(b/a), then ax – b is a factor. It’s a roundabout way of saying that if an expression divides evenly into a polynomial, then it follows that the expression is a factor.
One way to factor is to set the expression to equal 0, and then substitute various values of x until the equation is satisfied. Once you do that, you can determine that one of the factors is (x – whatever the number is. If it’s a negative, the expression would instead be x + the number. Subtracting a negative is the same as addition. Bear in mind that this is for only one factor.
The degree of an expression directly indicates how many factors it has. An expression leading with x^{2} has two factors.
If your polynomial contains a constant – that is, not a variable – you can sometimes factor it using that number. If the constant has no factors other than itself and one, this makes your job a little harder, but not impossible. The defining factor as to whether this is a solution is whether setting the expression equal to 0 results in a true statement.
To start, rewrite the expression as an equation that equals 0. Then, look at your constant. Start by taking the first factor of the constant, which is always going to be 1. For example, if you have the expression x3 – 3×2 – 10x + 24 = 0, you can assume that the factors are +-1, 2, 3, 4, 6, and 12.
Check both positive and negative results, because negative numbers require opposite signs and any of them could be the solution. Place each term in the equation one by one until you get as many true statements as possible:
You might be wondering about checking negative factors for a positive constant. In a cubic polynomial, this is impossible because you have three possible factors. Only three positives can result in a positive result.
When learning how to factor cubic polynomials, it helps to think of real-life applications. One of the most pervasive in modern life is the way our electricity functions. Alternating current, which powers our homes, constantly fluctuates in voltage and current. The function used for this is described as a sine wave, which is expressed as a polynomial.
Any type of curved function, such as the curve of a roller coaster, is expressed as a polynomial. Most often, you’ll have computers to help graph the difficult equations, but knowing the basics can help you understand the concepts.
The methods we’ve listed are just a few ways you can solve cubic polynomials. Note that not every expression can be factored. For example, if you have a polynomial with no solutions when you attempt to solve for zero, you can conclude that it never touches the x-axis. The graph of a cubic polynomial, however, may have three possible solutions, or two places where it curves.
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Entropy is the concept of chaos and disorder. It affects everything, including information. Learn how entropy works in calculus.
You’ve likely heard of the concept of entropy, or at least the word. Entropy has different specific definitions depending on the context in which you’re talking about it, but the common thread among the definitions is that of uncertainty and disorder. Entropy in any given system tends to increase over time when not provided with external energy or change.
Entropy, simply put, is the measure of disorder in a system. If you look at an ice cube floating in a mug of hot tea, you can think of it as an ordered system. The tea and the ice cube are separate. Over time, however, the heat from the tea transfers into the ice, causing it to melt and mix with the tea and leaving a homogenous mixture. There is no way to restore it to its previous state.
Everything in existence is subject to entropy per the second law of thermodynamics. There are several other definitions of the word, but what we’re here to talk about is the entropy information theory in calculus.
Information is data about a system, an object, or anything else in existence. Information is what differentiates one thing from another at the quantum level. The entropy of information has more to do with probability than with thermodynamics: simply put, the greater the number of possible outcomes of a system, the less able you are to learn new information.
Following from this, it also means that a low-data system has a high amount of entropy. For example, a six-sided die when cast has an equal chance (unless you’re using weighted dice) of landing on any one of the six sides; the exact percentage of probability for any side landing face-up is 16.67 percent.
A 12-sided die would have a probability of 8.33 percent for landing face-up on any of the twelve sides. Its information entropy is even higher than that of the six-sided die.
As another example, if you have a regular coin with two sides, flipping it yields a 50 percent probability of landing on either heads or tails. The coin, because there is a higher probability for either outcome, has a lower level of entropy than does the die.
The specific entropy information theory in calculus we’re talking about refers to data systems that have random outcomes, or at least some element of randomness. The level of entropy present in a data set refers to the amount of information you can expect to learn at any given time.
As such, the only factor that truly influences the level of entropy in a data set is the number of possible outcomes or the specificness of the information. For example, if you have a group of numbers from 1 to 10, and your only criteria is that the number is even, you only get one bit of information. It is measured in bits just as data in computers is.
The equation used for entropy information theory in calculus runs as such:
H = -∑n_{i=1} P(x_{i})log_{b}P(x_{i})
H is the variable used for entropy. The summation (Greek letter sigma), is taken between 1 and the number of possible outcomes of a system. For a 6-sided die, n would equal 6. The next variable, P(x_{i}), represents the probability of a given event happening. In this example, let’s say it represents the likelihood of the die landing face-up on 3. Other factors being equal, you get 1/6.
Then, you multiply this by the logarithm base b of P(x), where b is whichever base you’re using for your purposes. Most often, you’ll use 2, 10, or Euler’s number e. For reference, e is approximately equal to 2.71828.
This, of course, represents only a single discrete variable with no other factors influencing it. If you have another factor that can influence the outcome, the formula changes. Now, it ends up being the following:
H = -∑_{i,j} P(x_{i}, y_{j})log_{b}P(x_{i}, y_{j})/p(y_{j})
In this case, you have to look at both x and y as variables and take their functions as dependent on one another, depending on how the problem is set up. You can also simply treat the two variables as two separate and independent events that occur. For example, f you were to flip two coins, you could treat x and y as each coin coming up heads or tails.
Alternately, you could assume that you had one coin coming up heads, which would trigger another coin flip whereas landing on tails would not. Then, you’d take y as the variable representing the second coin. Again, it varies depending on what you’re doing.
You can determine how much information is received from an event with a reduction of the formula to
-∑p_{i}logp_{i}.
You might notice that the log has no indicated base. This is because by default the base is 10. In computer science, when charting out probabilities, you might see base 2 used a lot because of binary systems. Base e is used in many scientific disciplines. Base 10, meanwhile, is used in chemistry and other sciences that aren’t quite as heavy on math.
Base 10, after all, is the basis of the decimal system and the one most people are used to working with.
The above formula represents the average amount of information you can expect to gain from an event per iteration. It assumes all factors are equal and there are no influencing conditions on the outcome. In other words, it is completely random within the available set of data.
Entropy information theory in calculus has several possible measurements, depending on what base is being used for the logarithm. Here are a few of the most common measurements:
On occasion, you may have to convert one measurement to another.
One question that inevitably arises when dealing with higher mathematics is why it should be studied. After all, many skills in advanced math lack real-world applications, at least to the uninitiated. However, it should be noted that information and probability theory have several applications in computer science, such as file compression and text prediction.
For example, by using the entropy theory of information, you can help to code predictive text. The English language, for instance, has many different rules about what letters can occur in a sequence. If you see the letter ‘q’, you know that in all likelihood it’s followed the letter ‘u’. If you have two vowels, they will likely not be followed by a third, and certainly not by a fourth.
This is just one example. Another possible use in computer science is data compression as in ZIP files or image files like JPEGs. Information theory has also encompassed various scientific disciplines from statistics to physics. Even the inner workings of black holes, such as the presence of Hawking radiation, rely on information theory.
Even the act of learning higher math and abstract concepts can be beneficial, even without the knowledge. The act of learning forces the brain to process information in new ways, creating stronger connections between neurons and delaying the onset of cognitive decline. The more you learn, the more exercise your brain gets.
To get the most out of and understand information theory, you need a solid grasp on a few other subjects. These are all unsurprisingly math-related. Calculus and statistics are the two main subjects you need to study before starting to work on information theory because you’ll need to know how to take integrals and derivatives from calculus.
Statistics, meanwhile, gives you a solid understanding of probability theory and the likelihood of events occurring. You’ll also be introduced to a few of the more esoteric variables like lambda, or some of the symbols like sigma for summation if you haven’t seen them before.
You’ll also need to have a solid grasp of algebra, such as knowing how to manipulate equations and variables. Although you should get plenty of practice doing this in the course of algebra, it’s a good idea to understand some other concepts such as how to take a logarithm. As you can see, logarithms play a vital part in several of the entropy equations.
Learning the entropy information theory in calculus is a good way to understand how probability works and how many of the data systems you encounter produce various amounts of information. If you have a background in thermodynamic studies, it can make it easier to understand the concept of entropy.
Entropy in information theory is slightly different than it is in other branches of science, but the basic idea is the same.
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A logarithmic equation needs to be rewritten as an exponential equation for you to find the variable. This common calculus problem contains constants and expressions, and you’ll find logarithmic shortened to “log” in some written problems.
Condense a problem with more than one logarithm by turning it into one equation. A logarithm is constructed in a way that allows it to change a math function into another math function to solve a problem. Multiplication turns into addition and division becomes subtraction. The function changes let you find the variable value.
You’ll work with some well-known calculus rules with logs, such as:
The Power Rule
The Product Rule
The Quotient Rule
You will work with many components to solve logarithmic equations. Here are a few of them, with some brief information about the function of each one.
An exponent is a small raised number next to a larger numeral in an equation. The small number shows how many times the larger number is multiplied by itself. The small raised number 2 shows that the larger number is multiplied by itself twice. A number multiplied by itself twice can be referred to as 4 to the second power or four squared.
A variable is a symbol for an unknown number. Linear equations may have many values that can be used in place of the variable. Most variables, however, are solved with a single value.
The variable F represents the graph of a function. The graph of a function represents all points in f(x). You can also refer to the graph of a function as the graph of an equation.
When you look at the equation of a straight line (such as y equals mx plus b) the y-intercept is the location where the line goes through the vertical y-axis. In y equals mx plus b, b is the y-intercept, the slope is m, and the variable m is multiplied on the variable x.
An x-intercept appears at the spot where the graph crosses the x-axis. The x-intercept is also the point on the graph that shows the variable x as zero.
A problem with one logarithm on each side of the equation that has the same base lets you use arguments that are the same. The expressions M and N are the arguments in the following description:
Log_{b }M equals Log_{b }N leads to M equals N
A problem with a logarithm on one side of the equation you can use an exponential equation to find the answer.
Log_{b} M equals N leads to M equals b^{N}
Solve this problem:
Log_{3 }(x) plus Log_{3 }(x -2) equals Log_{3} (x plus 10)
Condense the log arguments on the left side into one logarithm with the Product Rule. You need one log expression on both sides of the equation. X will have a power of two, so you’ll need to solve a quadratic equation.
You get Log_{3 }[(x) (x minus 2)] equals Log_{3 }(x plus 10). Now condense (x) (x-2) = x squared minus 2x. Then you get Log_{3 }(x squared minus 2x) equals Log_{3 }(x plus 10)
Get rid of the logs and set the arguments inside the parenthesis to match each other.
X squared minus 2x equals x plus 10
Now use the factoring method to finish the quadratic equation. Move all information to one side and make the opposite side contain a zero value.
X squared minus 3x minus 10 equals zero
(x minus 5) (x plus 2) equals zero
Now turn each factor to zero and solve x. X minus 5 equals zero means that x equals 5. X plus 2 equals zero means that x equals negative 2. X equals 5 and X equals negative 2 are the answers we may use. Now check the answers to see if they are correct.
Place the answers back in the first logarithmic equation to verify their validity.
For x equals 5, Log_{3 }(5) plus Log_{3} (5-2) equals Log_{3 }(5 plus 10)
Log_{3 }(5) plus Log_{3 }(3) equals Log_{3 }(15)
This is the correct answer. X equals negative 2 gives us a few negative numbers inside the parenthesis, and a log of zero and negative numbers in the equation make negative 2 the wrong answer.
Solve the following problem: ½ log (X to the fourth power) minus log (2x minus 1) equals log (x squared) plus log (2).
Log without a written base has a base of 10. Base 10 is the common logarithm. Compress both sides of the equation into one log. You’ll see the Quotient Rule applied on the left side ( a difference of logs) and the Product Rule (the sum of logs) on the right side.
Pay attention to the ½ coefficient on the left side. You’ll need to use the Power Rule and bring the coefficient ½ up in reverse order.
Log (base), M to the k power equals K times log (base) M, then ½ log (X to the fourth power) minus log (2x minus 1) equals log (x squared) plus log (2)
Now use the ½ as an exponent on the left. Log (x to the fourth power) to the one/half power minus log (2x minus 1) equals log (x to the second power) plus log (2).
Now simplify the exponent to log (x squared) minus log (2x minus 1) equals log (x squared) plus log (2) Now condense log using the Product Rule on the right and the Quotient Rule on the left.
Log ( x squared over 2x minus one) equals log (2x to the second power)
It’s all right to show that if we have the same base in our equations (base 10), we can show that they are equal to each other. Now drop the logs and put arguments inside their parentheses.
X squared over 2x minus 1 equals 2x squared
Use the Cross Product to solve the Rational Equation. Factor out to get the brief, final answer after moving all terms to one side of the equation. Make each factor equal to zero and then solve x. X equal ¾ is one possible answer, x equals zero is the other.
Check your possible answers. X equals zero bring an undefined zero logarithm into the equation, which is wrong. X equals 3.4 is the only solution.
Students who consistently get excellent grades in calculus or geometry do so because they study daily and show a steady interest in their classwork and homework. Establish a study routine by choosing a quiet place where you can read and practice solving problems at the same time each day (or at least a few times a week).
Bookmark math websites and videos containing videos that will help you understand the formulas and concepts that give you trouble, including any logarithmic equation. Along with your class notes and practice problems in your textbook, you’ll be equipped with everything you need to study more efficiently.
Contact other students in your class, tutors from the math department, or independent math tutors to help you if you’re unable to master a particular formula on your own. Even the best students need help from another student or a tutor from time to time.
Don’t expect calculus or any math class to be difficult or easy; work on the assignments without worrying about your grades or the outcome. Study for tests up to a week in advance, preferably with classmates or other math students. Trade tips and discuss different solutions and approaches to problems.
No one fails calculus because they lack the skills or mental capacity to perform exercises properly. People fail because they are unwilling or unable to do the required work. Teachers and calculus experts suggest you study six or seven hours on weekends and a few hours each weeknight to get the best grades possible.
Anyone who only has five to ten hours a week to study and prepare for class or tests should delay calculus classes until they are ready to study more often. Calculus courses are fast-paced, and if you get behind it will be hard to get caught up on the lessons you missed or didn’t understand.
Work as hard as you can in the first month of class. Define your strengths and weaknesses, and enlist tutors or a study group if you need them. Don’t get behind and think you can catch up quickly without help. You may end up dropping the class if you don’t take charge of your studies.
Don’t miss classes. Calculus isn’t like history or English, where catching up is hard, but not impossible, by yourself. If you miss one class, you should be able to catch up if you have a passable grasp of previous classes.
Anyone with a poor understanding of previous formulas and problem-solving methods will find that they are completely lost after missing a class or two. As soon as you feel confused, let your teacher, tutor, or study group know and ask for help.
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Figuring out how to find the area a under curve in a graph can be a simple process once you understand the formula and the surrounding pieces of information given in the problem. Finding the area is part of integration mathematics, and by using the appropriate formula, we can calculate not just the area, but any given quantity.
A typical graph has an x-axis and a y-axis, and when you add a curve to this structure, you’ll immediately see where the area under the curve lies. By finding the points along the curve, we can input them into the formula for finding the area under a curve, and solve for the final answer.
Calculus is one kind of mathematics and deals with finding different properties of integrals of functions and other derivatives. To see these values, specific mathematical methods are used that are based on the sums of differences which are infinitesimal.
There are two primary types of calculus, which are integral calculus and differential calculus. Overall, calculus can be summed up as a system of calculating and reasoning different values as they undergo constant change.
These two kinds of calculus are connected to each other by a theory called the fundamental theorem of calculus. Both kinds of calculus use notions such as infinite sequences, convergence, and infinite series as it applies to limits that are well defined.
The calculus that is taught currently is credited to the 17th-century mathematicians Isaac Newton and Gottfried Leibniz. Calculus today is now widely used in other disciplines such as economics, engineering, and science.
Calculus is now part of the modern curriculum taught in most schools, and it can serve as a gateway to more complicated mathematical learning, and other studies regarding limits, functions, and additional mathematical analysis.
It’s possible that you may have also heard calculus referred to by other names such as infinitesimal calculus, or “the calculus of infinitesimals. Calculus has also been employed as a name for different kinds of mathematical notation such as Ricci calculus, lambda calculus, propositional calculus, process calculus, and calculus of variations.
Calculus dates back to ancient and even medieval times where it was used to develop different ideas surrounding calculations such as area, volume, and early ideas about limits. Calculus has been used to discover different values such as the area of a circle, and the volume of a sphere.
There are many different real-world applications where finding the area under a curve can be useful, and many of these don’t have anything to do with math alone. The formulas and method for finding the area under a curve can be useful in a variety of subjects including:
Biology
Statistics
Medicine